Woven Graph Codes:
Asymptotic Performances and Examples

In order to illustrate the structure of a binary graph-based block code with constituent block codes we represent the Heawood bipartite graph using a so-called Tanner graph [15] as shown in Fig. 4.

We introduce a set of $nc=21$ (variable) vertices which correspond to the code symbols. Each of the $2n=14$ (constraint) vertices on the right- and left-hand sides corresponds to one of 14 parity checks. The $c=3$ edges leaving one constraint vertex correspond to a codeword of the constituent $(c,b)$ block code of rate $R^{c}=b/c$. The parity-check matrix of the corresponding graph-based code with binary constituent block codes is

$$H_{\rm gb}=\begin{pmatrix}H_{1}\\ H_{2}\end{pmatrix}$$

(1)

where the parity-check matrix $H_{1}$ of size $n\times nc=7\times 21$ has the form

$$H_{1}=\left(\begin{array}[]{ccccccc}H^{c}&{\boldsymbol{0}}&{\boldsymbol{0}}&{\boldsymbol{0}}&{\boldsymbol{0}}&{\boldsymbol{0}}&{\boldsymbol{0}}\\ {\boldsymbol{0}}&H^{c}&{\boldsymbol{0}}&{\boldsymbol{0}}&{\boldsymbol{0}}&{\boldsymbol{0}}&{\boldsymbol{0}}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ {\boldsymbol{0}}&{\boldsymbol{0}}&{\boldsymbol{0}}&{\boldsymbol{0}}&{\boldsymbol{0}}&{\boldsymbol{0}}&H^{c}\end{array}\right)$$

where $H^{c}$ is a size $(c-b)\times c=(3-b)\times 3$ parity-check matrix of the constituent block code, and $H_{2}$ is a size $n\times nc=7\times 21$ parity-check matrix which is the permutation of the columns of $H_{1}$ determined by the graph. Notice that in general by choosing $b<c$ and assigning constituent block codes of different rates $R^{c}=b/c$ to the same graph we can obtain graph-based codes of different rates. In general, since in an $s$-partite, $s$-uniform, $c$-regular hypergraph the total number of parity checks is equal to $sn(c-b)$, the code rate $R_{\rm gb}$ of the graph-based code is

$$R_{\rm gb}\geq\frac{n(c-s(c-b))}{nc}=s(R^{c}-1)+1$$

(2)

with equality if and only if all parity-checks are linearly independent. If $s=2$, then we get $R_{\rm gb}\geq 2R^{c}-1$.

The simplest example of a Heawood graph-based code can be obtained by choosing as constituent block codes a single-parity-check code of rate $R^{c}=1/3$. Then the parity-check matrix $H^{c}$ has the form

$$H^{c}=\left(\begin{array}[]{lll}1&1&1\end{array}\right)$$

and the parity-check matrix of the graph-based code is

$$\footnotesize H_{\rm gb}=H_{\rm g}=\left(\begin{array}[]{ccc:ccc:ccc:ccc:ccc:ccc:ccc|r}1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19&20&21&\\ \hline\cr 1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&2\\ 0&0&0&0&0&0&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&4\\ 0&0&0&0&0&0&0&0&0&1&1&1&0&0&0&0&0&0&0&0&0&6\\ 0&0&0&0&0&0&0&0&0&0&0&0&1&1&1&0&0&0&0&0&0&8\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&1&1&0&0&0&10\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&1&1&12\\ \hline\cr 1&0&0&0&1&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&1\\ 0&0&0&1&0&0&0&1&0&0&0&0&0&0&1&0&0&0&0&0&0&3\\ 0&0&0&0&0&0&1&0&0&0&1&0&0&0&0&0&0&1&0&0&0&5\\ 0&0&0&0&0&0&0&0&0&1&0&0&0&1&0&0&0&0&0&0&1&7\\ 0&0&1&0&0&0&0&0&0&0&0&0&1&0&0&0&1&0&0&0&0&9\\ 0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&1&0&0&0&1&0&11\\ 0&1&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&1&0&0&13\end{array}\right).$$

(3)

In this case the graph-based code coincides with the graph code since (3) is the incidence matrix of the Heawood graph. In [4] it is proved that the minimum distance of the bipartite graph-based code with single-parity-check constituent codes is $d_{\min}=g$, where $g$ is the girth of the corresponding graph. Notice that for the Tanner graph we have $d_{\min}=g/2$. The parity-check matrix (3) is a $14\times 21$ parity-check matrix. Taking into account that one check is linearly dependent on the other, we obtain a $(21,8)$ binary block code. Its minimum distance is $d_{\min}=g=6$.

Consider the hypergraph shown in Fig. 1. Its incidence matrix has the form

$$\footnotesize H_{\rm hg}=H_{\rm hgb}=\left(\begin{array}[]{cccc:cccc:cccc:cccc|r}1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&\\ \hline\cr 1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&1&1&1&1&0&0&0&0&0&0&0&0&1\\ 0&0&0&0&0&0&0&0&1&1&1&1&0&0&0&0&2\\ 0&0&0&0&0&0&0&0&0&0&0&0&1&1&1&1&3\\ \hline\cr 1&0&0&0&0&1&0&0&0&0&1&0&0&0&0&1&4\\ 0&1&0&0&0&0&1&0&0&0&0&1&1&0&0&0&5\\ 0&0&1&0&0&0&0&1&1&0&0&0&0&1&0&0&6\\ 0&0&0&1&1&0&0&0&0&1&0&0&0&0&1&0&7\\ \hline\cr 1&0&0&0&1&0&0&0&1&0&0&0&1&0&0&0&8\\ 0&1&0&0&0&1&0&0&0&1&0&0&0&1&0&0&9\\ 0&0&1&0&0&0&1&0&0&0&0&1&0&0&0&1&10\\ 0&0&0&1&0&0&0&1&0&0&1&0&0&0&1&0&11\\ \end{array}\right)$$

(4)

and is a $12\times 16$ parity-check matrix of a hypergraph-based code which coincides with the parity-check matrix of the hypergraph code. Each column represents a hyperedge and each row represents a vertex of this hypergraph. For example, the first four rows represent the vertices 1, 2, 3, and 4 (triangles), the next four rows he vertices 5, 6, 7, and 8 (rectangles), and the last four rows the vertices 9, 10, 11, and 12 (ovals). The first column represents the hyperedge which connects the vertices 1, 5, and 9, the second column the hyperedge connecting vertices 1, 6, and 10 etc. The rows of (4) are linearly dependent. By removing two parity checks we obtain a $(16,6)$ linear block code with the minimum distance $d_{\min}=g_{3,2}=6$, where $g_{3,2}$ is the ($3$,$2$)-girth of the hypergraph. The rate of this hypergraph code is $R_{\rm hg}=3/8$, which satisfies inequality (2),

$$R_{\rm hg}\geq 3\left(\frac{3}{4}-1\right)+1=\frac{1}{4}.$$

The Tanner version of this hypergraph is shown in Fig. 5.

For an $s$-partite, $s$-uniform, $c$-regular hypergraph-based code with constituent block codes we have the following theorem.

Proof. Any nonzero codeword in an $s$-partite, $s$-uniform, $c$-regular hypergraph-based code always corresponds to a connected $(\geq d_{\min}^{c})$-subgraph or a set of disjoint connected subgraphs. These subgraphs are called active [14], [4]. All hyperedges and vertices in an active subgraph are also called active. The number of hyperedges in the shortest connected subgraph is equal to $g_{s,d_{\min}^{c}}$. Any nonzero symbol in a codeword corresponds to an active hyperedge in the graph. By using the arguments given above, we conclude that for any codeword ${\boldsymbol{v}}$,

$$w_{\rm H}({\boldsymbol{v}})\geq g_{s,d_{\min}^{c}}$$

where $w_{\rm H}({\boldsymbol{v}})$ is the Hamming weight of ${\boldsymbol{v}}$. Minimizing over ${\boldsymbol{v}}$ completes the proof.

Now assume that the constituent code assigned to the hypergraph vertices is a binary $(lc,lb)$ linear block code determined by a parity-check matrix

$$H^{c}=\left(\begin{array}[]{llll}H^{c}_{11}&H^{c}_{12}&\dots&H^{c}_{1,c}\\ H^{c}_{21}&H^{c}_{22}&\dots&H^{c}_{2,c}\\ \vdots&\vdots&\ddots&\vdots\\ H^{c}_{(c-b),1}&H^{c}_{(c-b),2}&\dots&H^{c}_{(c-b),c}\end{array}\right)$$

(5)

where $H^{c}_{ij}\in{\mathcal{B}}_{l\times l}$ is a size $l\times l$ matrix, ${\mathcal{B}}_{l\times l}$ is the set of all possible binary matrices of size $l\times l$.

Let ${\mathcal{C}}_{2}(H^{c})$ denote such a binary $(lc,lb)$ constituent block code determined by the matrix (5). We call the corresponding hypergraph-based code with ${\mathcal{C}}_{2}(H^{c})$ as constituent codes a woven graph code with constituent block codes.

Consider an example of a woven graph code based on the bipartite graph with girth $g=4$ shown in Fig. 6. The Tanner version of this the so-called “utility” bipartite graph is shown in Fig. 7.

The incidence matrix of this graph is

$$H_{\rm g}=\left(\begin{array}[]{ccc:ccc:ccc}1&1&1&0&0&0&0&0&0\\ 0&0&0&1&1&1&0&0&0\\ 0&0&0&0&0&0&1&1&1\\ \hline\cr 1&0&0&0&1&0&0&0&1\\ 0&0&1&1&0&0&0&1&0\\ 0&1&0&0&0&1&1&0&0\end{array}\right).$$

(6)

We use a constituent ($4\times 3,4\times 2$) linear block code with $d_{\min}^{c}=3$ determined by the parity-check matrix

	$\displaystyle H^{c}$	$\displaystyle=$	$\displaystyle\left(H_{1}^{c}\quad H_{2}^{c}\quad H_{3}^{c}\right)$
		$\displaystyle=$	$\displaystyle\left(\begin{array}[]{cccc:cccc:cccc}1&0&0&0&1&1&1&0&1&1&0&0\\ 0&1&0&0&0&1&1&1&0&1&1&0\\ 0&0&1&0&1&0&1&1&0&0&1&1\\ 0&0&0&1&1&1&0&1&1&0&0&1\\ \end{array}\right).$

By searching over all possible permutations of the matrices $H_{1}^{c}$, $H_{2}^{c}$, and $H_{3}^{c}$ we found the following parity-check matrix of the woven graph code with the best minimum distance

$$\footnotesize H_{\rm wg}=\left(\begin{array}[]{ccc:ccc:ccc}H_{1}^{c}&H_{2}^{c}&H_{3}^{c}&\bf 0&\bf 0&\bf 0&\bf 0&\bf 0&\bf 0\\ \bf 0&\bf 0&\bf 0&H_{1}^{c}&H_{2}^{c}&H_{3}^{c}&\bf 0&\bf 0&\bf 0\\ \bf 0&\bf 0&\bf 0&\bf 0&\bf 0&\bf 0&H_{1}^{c}&H_{2}^{c}&H_{3}^{c}\\ \hline\cr H_{2}^{c}&\bf 0&\bf 0&\bf 0&H_{3}^{c}&\bf 0&\bf 0&\bf 0&H_{1}^{c}\\ \bf 0&\bf 0&H_{1}^{c}&H_{2}^{c}&\bf 0&\bf 0&\bf 0&H_{3}^{c}&\bf 0\\ \bf 0&H_{3}^{c}&\bf 0&\bf 0&\bf 0&H_{1}^{c}&H_{2}^{c}&\bf 0&\bf 0\\ \end{array}\right).$$

(8)

The matrix (8) describes a $(36,12)$ linear block code with $d_{\min}=10$.

Any codeword ${\boldsymbol{v}}^{c}$ of the $(lc,lb)$ constituent block code can be represented as a sequence of $c$ blocks of length $l$, that is, ${\boldsymbol{v}^{c}}=({\boldsymbol{v}}_{1}^{c},{\boldsymbol{v}}_{2}^{c},\dots,{\boldsymbol{v}}^{c}_{c})$, where ${\boldsymbol{v}}_{i}^{c}=(v^{c}_{i1},v^{c}_{i2},\dots,v^{c}_{il})$, $i=1,2,\dots,c$. We define the minimum Hamming block distance between the codewords ${\boldsymbol{v}}^{c}$ and ${\boldsymbol{\tilde{v}}}^{c}$ of the constituent block code as

$$d_{block}^{c}=\min_{{\boldsymbol{v}}^{c}\neq{\boldsymbol{\tilde{v}}}^{c}}\left\{w_{block}({\boldsymbol{v}}^{c}-{\boldsymbol{\tilde{v}}}^{c})\right\}$$

where $w_{block}^{c}({\boldsymbol{v}}^{c})=\#({\boldsymbol{v}}_{i}^{c}\neq{\boldsymbol{0}})$, $i=1,2,\dots,c$. Next we will prove the following theorem.

Proof. Any nonzero codeword corresponds to an active connected subgraph or a set of disjoint connected subgraphs and the number of hyperedges in the shortest subgraph is $g_{s,d_{block}^{c}}$. Any nonzero symbol in a codeword activates a hyperedge in the graph, that is, not less than $s$ constituent subcodes correspond to a codeword. Since at most $c$ hyperedges are connected with any hypergraph vertex then the number of active constituent subcodes can be lower-bounded by

$$\frac{g_{s,d_{block}^{c}}}{c}.$$

Taking into account that any codeword of block weight greater than or equal to $d_{block}^{c}$ in the constituent block code has a weight at least equal to $d_{\min}^{c}$ we obtain the following inequality

$$w_{\rm H}({\boldsymbol{v}})\geq\max\left\{\frac{g_{s,d_{block}^{c}}}{c},s\right\}d_{\min}^{c}$$

for any codeword ${\boldsymbol{v}}$ and the proof is complete.

From Theorem 2 for the woven graph code determined by (8) we obtain that $d_{\min}\geq\max\left\{\frac{4}{4},3\right\}3=9.$

Woven graph codes with constituent convolutional codes can be considered as a straightforward generalization of the woven graph code with constituent block codes. Assume that the ${\mathcal{C}}_{2}(H^{c})$ code is chosen as a zero-tail terminated (ZT) convolutional code and consider a sequence of ZT convolutional codes with increasing $l$. It is evident that when $l$ tends to infinity the $(lc,lb)$ constituent code ${\mathcal{C}}_{2}(H^{c})$ can be chosen as a rate $R^{c}=b/c$ binary convolutional code with constraint length $\nu^{c}$. Then the corresponding woven graph code has rate $R=s(R^{c}-1)+1$ and its constraint length is at most $sn\nu^{c}$.

Another description of woven graph codes with constituent convolutional codes follows from the representation of the constituent convolutional code in polynomial form. Let $G^{c}(D)$ be a minimal encoding matrix [8] of a rate $R^{c}=b/c$, memory $m^{c}$ convolutional code, given in polynomial form, that is,

$$\vskip 5.69054ptG^{c}(D)=\begin{pmatrix}g^{c}_{11}(D)&\dots&g^{c}_{1c}(D)\\ \vdots&\ddots&\vdots\\ g^{c}_{b1}(D)&\dots&g^{c}_{bc}(D)\end{pmatrix}$$

(9)

where $g^{c}_{ij}(D)=g_{ij}^{c(0)}+g_{ij}^{c(1)}D+g_{ij}^{c(2)}D^{2}+\dots+g_{ij}^{c(m)}D^{m}$, $i=1,2,\dots,b$, $j=1,2,\dots,c$, are binary polynomials such that $m^{c}=\max_{i,j}\{\deg g^{c}_{ij}(D)\}$. The overall constraint length is $\nu^{c}=\sum_{i}\max_{j}\{\deg g^{c}_{ij}(D)\}$. The binary information sequence ${\boldsymbol{u}}^{c}(D)=(u_{1}^{c}(D),u_{2}^{c}(D),\dots,u_{b}^{c}(D))$ is encoded as

$${\boldsymbol{v}}^{c}(D)={\boldsymbol{u}}^{c}(D)G^{c}(D)$$

where ${\boldsymbol{v}}^{c}(D)=(v_{1}^{c}(D),v_{2}^{c}(D),\dots,v_{c}^{c}(D))$ is a binary code sequence. Let $H^{c}(D)$ denote a parity-check matrix for the same code,

$$H^{c}(D)=\begin{pmatrix}h_{11}^{c}(D)&\dots&h_{1c}^{c}(D)\\ \vdots&\ddots&\vdots\\ h_{r1}^{c}(D)&\dots&h_{rc}^{c}(D)\end{pmatrix}$$

(10)

where $r=c-b$ is the redundancy of the constituent code.

We denote by ${\mathds{F}}_{2}((D))$ the field of binary Laurent series and regard a rate $R^{c}=b/c$ constituent convolutional code as a rate $R^{c}=b/c$ block code ${\mathcal{C}}^{c}$ over the field of binary Laurent series encoded by $G^{c}(D)$. Then its codewords ${\boldsymbol{v}}^{c}(D)$ are elements of ${\mathds{F}}_{2}((D))^{c}$, which is the $c$-dimensional vector space over the field of binary Laurent series [8].

The minimum Hamming block distance between the codewords ${\boldsymbol{v}}_{j}(D)$ and ${\boldsymbol{v}}_{k}(D)$ is defined [9] as

$$d_{\rm block}=\min_{{\boldsymbol{v}}_{j}(D)\neq{\boldsymbol{v}}_{k}(D)}\{w_{\rm block}({\boldsymbol{v}}_{j}(D)-{\boldsymbol{v}}_{k}(D))\}$$

Representing a convolutional code as a block code over the field of binary Laurent series we can obtain a woven graph code with constituent convolutional codes as a generalization of a graph-based code with binary constituent block codes. For example, a parity-check matrix $H_{\rm wg}(D)$ of the rate $R_{\rm wg}=4/3-1=1/3$ Heawood’s graph-based code with $R^{c}=2/3$ constituent convolutional codes has the form

$$H_{\rm wg}(D)=\footnotesize\left(\begin{array}[]{ccc:ccc:ccc:ccc:ccc:ccc:ccc}h^{c}_{1}&h^{c}_{2}&h^{c}_{3}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&h^{c}_{1}&h^{c}_{2}&h^{c}_{3}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&h^{c}_{1}&h^{c}_{2}&h^{c}_{3}&0&0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&h^{c}_{1}&h^{c}_{2}&h^{c}_{3}&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0&0&h^{c}_{1}&h^{c}_{2}&h^{c}_{3}&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&h^{c}_{1}&h^{c}_{2}&h^{c}_{3}&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&h^{c}_{1}&h^{c}_{2}&{\boldsymbol{h}}^{c}_{3}\\ \hline\cr t^{c}_{1}&0&0&0&t^{c}_{3}&0&0&0&0&0&0&t^{c}_{2}&0&0&0&0&0&0&0&0&0\\ 0&0&0&t^{c}_{1}&0&0&0&t^{c}_{3}&0&0&0&0&0&0&t^{c}_{2}&0&0&0&0&0&0\\ 0&0&0&0&0&0&t^{c}_{1}&0&0&0&t^{c}_{3}&0&0&0&0&0&0&t^{c}_{2}&0&0&0\\ 0&0&0&0&0&0&0&0&0&t^{c}_{1}&0&0&0&t^{c}_{3}&0&0&0&0&0&0&t^{c}_{2}\\ 0&0&t^{c}_{2}&0&0&0&0&0&0&0&0&0&t^{c}_{1}&0&0&0&t^{c}_{3}&0&0&0&0\\ 0&0&0&0&0&t^{c}_{2}&0&0&0&0&0&0&0&0&0&t^{c}_{1}&0&0&0&t^{c}_{3}&0\\ 0&t^{c}_{3}&0&0&0&0&0&0&t^{c}_{2}&0&0&0&0&0&0&0&0&0&t^{c}_{1}&0&0\end{array}\right)$$

(11)

where $h^{c}_{i}$ and $t^{c}_{i}$ are short-hand for $h^{c}_{i}(D)$ and $t^{c}_{i}(D)$, respectively, and $H^{c}(D)=\left(h_{1}^{c}(D)\quad h_{2}^{c}(D)\quad h_{3}^{c}(D)\right)$ is a parity-check matrix of the rate $R^{c}=2/3$ constituent convolutional code and $(t^{c}_{1}(D),t^{c}_{2}(D),t^{c}_{3}(D))$ is one of six possible permutations of $h_{1}^{c}(D),h^{c}_{2}(D),h^{c}_{3}(D)$.

Exploiting the above definitions we can interpret this bipartite woven graph-based code with constituent convolutional codes as follows. The left column of vertices in Fig. 4 represents $n$ parity checks each of which determines one of $n$ constituent fixed and identical convolutional codes and their $nc$ branches represent the elements $v_{ij}^{c\mathrm{L}}(D)\in{\mathds{F}}_{2}((D))$, $i$ even, $0\leq i\leq 2n-2$, $1\leq j\leq c$. Similarly, the right column of vertices represents same convolutional codes and their $nc$ branches represent the elements $v_{ij}^{c\mathrm{R}}(D)\in{\mathds{F}}_{2}((D))$, $i$ odd, $1\leq i\leq 2n-1$, $1\leq j\leq c$, where the set $\{v_{ij}^{c\mathrm{R}}(D)\}$ is a random permutation of the set $\{v_{ij}^{c\mathrm{L}}(D)\}$ determined by the graph.

We can also regard the $n$ left constituent convolutional codes as a warp with $nc$ threads. Each of the $n$ right constituent convolutional codes are tacked on $c$ of the threads in the warp such that each thread of the warp is tacked on exactly once. Thus, our construction is a special case of a woven code [17] and we call this graph-based code a woven graph code.

Proof. Since woven graph codes with constituent convolutional codes can be considered as a generalization of woven graph codes with constituent block codes, the theorem follows from Theorem 2 when $l$ tends to infinity.

For a woven graph code based on a bipartite graph with girth $g$ and containing constituent convolutional codes with minimum block distance $d_{block}^{c}=2$ and free distance $d_{\rm free}^{c}$ by a straightforward generalization of the approach of [4] we obtain the following tighter bound on the free distance

$$d_{{\rm free}}\geq\max\left\{\frac{g}{2},2\right\}d^{c}_{\rm free}.$$

(12)

First we consider the ensemble of random woven graph codes with rate $R^{c}=b/c$ constituent block codes determined by the edges of a random $s$-partite, $s$-uniform, $c$-regular hypergraph corresponding to the time-varying random parity-check matrix

$$H_{\rm wg}=\begin{pmatrix}{\tilde{H}}_{1}\\ {\tilde{H}}_{2}\\ \vdots\\ {\tilde{H}}_{s}\end{pmatrix}=\begin{pmatrix}\pi_{1}(H_{1})\\ \pi_{2}(H_{2})\\ \vdots\\ \pi_{s}(H_{s})\end{pmatrix}$$

(13)

where ${\tilde{H}}_{i}=\pi_{i}(H_{i})$, $i=1,2,\dots,s$, is a block matrix of size $nc(1-R^{c})\times nc$ (or a binary matrix of size $n(c-b)l\times ncl$) and $\pi_{i}$ denotes a random permutation of the columns of $H_{i}$,

$$H_{i}=\begin{pmatrix}H_{i}^{c(1)}&\boldsymbol{0}&\dots&\boldsymbol{0}\\ \boldsymbol{0}&H_{i}^{c(2)}&\boldsymbol{0}&\dots\\ \vdots&\dots&\ddots&\vdots\\ \boldsymbol{0}&\dots&\boldsymbol{0}&H_{i}^{c(n)}\end{pmatrix}$$

(14)

where $H_{i}^{c(t)}$, $t=1,\dots,n$, denotes the random parity-check matrix (5) which determines the $(lc,lb)$ constituent block code and $n$ is the number of constituent codes in each subgraph.

Remark: In [3] a more restricted ensemble of random codes is studied in which all matrices are identical random matrices. In the proof of Theorem 1 we need that the syndrome components are independent random variables in the product probability space of random matrices and random permutations. The following simple example shows that this is not always the case if all matrices are identical.

Consider $n=1$ constituent block codes of block length $c=2$ with $b=1$ information symbols. This example is rather artificial since the rate of the constituent block code $R^{c}=1/2$ and therefore the rate of the graph-based code with $s=2$ is $R_{\rm wg}=s(R^{c}-1)+1=2R^{c}-1=0$. In this case the parity-check matrix of the code has the form

$$H_{\rm wg}=\left(\begin{array}[]{c}H_{1}\\ \pi(H_{2})\\ \end{array}\right)$$

where $\pi$ is a random permutation of $c$ elements. First assume that all matrices are identical, that is, $H_{1}=H_{2}$. There are only 8 equiprobable elements in the product space, namely,

	$\displaystyle\{H_{\rm wg}\}$	$\displaystyle=$	$\displaystyle\left\{\left(\begin{array}[]{cc}0&0\\ 0&0\\ \end{array}\right),\left(\begin{array}[]{cc}0&0\\ 0&0\\ \end{array}\right),\left(\begin{array}[]{cc}0&1\\ 0&1\\ \end{array}\right),\left(\begin{array}[]{cc}0&1\\ 1&0\\ \end{array}\right),\right.$
			$\displaystyle\left.\left(\begin{array}[]{cc}1&0\\ 1&0\\ \end{array}\right),\left(\begin{array}[]{cc}1&0\\ 0&1\\ \end{array}\right),\left(\begin{array}[]{cc}1&1\\ 1&1\\ \end{array}\right),\left(\begin{array}[]{cc}1&1\\ 1&1\\ \end{array}\right)\right\}.$

For any vector ${\boldsymbol{x}}$ of weight 1 we have the following set of random equiprobable syndromes:

	$\displaystyle\{{\boldsymbol{x}}H_{\rm wg}^{\rm T}\}$	$\displaystyle=$	$\displaystyle\left\{\left(\begin{array}[]{cc}0&0\\ \end{array}\right),\left(\begin{array}[]{cc}0&0\\ \end{array}\right),\left(\begin{array}[]{cc}0&0\\ \end{array}\right),\left(\begin{array}[]{cc}0&1\\ \end{array}\right),\right.$
			$\displaystyle\left.\left(\begin{array}[]{cc}1&1\\ \end{array}\right),\left(\begin{array}[]{cc}1&0\\ \end{array}\right),\left(\begin{array}[]{cc}1&1\\ \end{array}\right),\left(\begin{array}[]{cc}1&1\\ \end{array}\right)\right\}.$

Therefore,

$$P({\boldsymbol{x}}H_{\rm wg}^{\rm T}={\boldsymbol{0}}|w_{\rm H}({\boldsymbol{x}})=1)=\frac{3}{8}>\frac{1}{4}.$$

If $H_{1}$ and $H_{2}$ are both random and independent this probability is equal to 1/4.

Although this remark contradicts the proof of Theorem 3 in [3], there exists another (combinatorial) way to prove the same statement for identical $H_{i}$ [16].

Next we prove the following theorem.

where $\delta(R_{\rm wg})$ is a root of the equation

$$(1-s)h(\delta)-\delta s\log_{2}\left(2^{-(R_{\rm wg}-1)/s}-1\right)=0$$

and $\delta_{\rm VG}(R_{\rm wg})$ is the solution of $h(\delta)+R_{\rm wg}-1=0$, and $h(\cdot)$ denotes the binary entropy function.

Proof. Let $w$ be the Hamming weight of the codeword ${\boldsymbol{v}}$ of the random binary woven graph code ${\mathcal{C}}_{2}(H_{\rm wg})$. We are going to find a parameter $d$ such that the probability ${\rm P}({\boldsymbol{v}}H_{\rm wg}^{\mathrm{T}}={\boldsymbol{0}}|w)$ tends to 0 for all $w<d$. We can rewrite ${\rm P}({\boldsymbol{v}}H_{\rm wg}^{\mathrm{T}}={\boldsymbol{0}}|w)$ as

$${\rm P}({\boldsymbol{v}}H_{\rm wg}^{\mathrm{T}}={\boldsymbol{0}}|w)=\sum_{{\boldsymbol{j}}}{\rm P}({\boldsymbol{v}}H_{\rm wg}^{\mathrm{T}}={\boldsymbol{0}}|w,{\boldsymbol{j}}){\rm P}({\boldsymbol{j}}|w)$$

(20)

where ${\boldsymbol{j}}=(j_{1},j_{2},\dots,j_{s})$ and $j_{i}$ denotes the number of nonzero constituent codewords in the $i$th subgraph corresponding to the codeword of weight $w$.

In the ensemble of random parity-check matrices $H^{c(t)}$, $t=1,2,\dots,n$, of size $lc(1-R^{c})\times lc$ the probability that a nonzero vector $\boldsymbol{v}^{c}$ is a codeword of the corresponding constituent random binary code ${\mathcal{C}}_{2}(H^{c})$ is equal to $2^{-(c-b)l}$ since the syndromes of the constituent codes are equiprobable sequences of length $(c-b)l$. Taking into account that in the $i$th subgraph we have $j_{i}$ nonzero constituent codewords the probability ${\rm P}({\boldsymbol{v}}H_{\rm wg}^{\mathrm{T}}={\boldsymbol{0}}|w,{\boldsymbol{j}})$ can be upper-bounded by

$${\rm P}({\boldsymbol{v}}H_{\rm wg}^{\mathrm{T}}={\boldsymbol{0}}|w,{\boldsymbol{j}})\leq\binom{ncl}{w}\prod_{i=1}^{s}2^{-j_{i}cl(1-R^{c})}.$$

(21)